Final answer:
An improper integral is characterized by having infinite limits or an integrand that becomes infinite within the integration interval. To determine the values that make an integral improper, check for infinite integration limits or points where the function is undefined or approaches infinity.
Step-by-step explanation:
An improper integral is an integral that has either infinite limits of integration or an integrand that approaches infinity at some points within the limits of integration. To determine the values that make an integral improper, one must look for any of the following conditions:
- The limits of integration include infinity or negative infinity.
- The function being integrated, or the integrand, becomes infinite (or undefined) at some point within the intervals of integration.
For example, if you have an integral such as ∫∞a f(x) dx, where a is finite, the integral is improper because one of the limits of integration is infinite. Similarly, if the integral of a function f(x) between a and b includes a point c where f(c) is undefined or goes to infinity, then the integral ∫ba f(x) dx is also improper.
To evaluate an improper integral, one typically uses limits to define the integral around the problematic value(s) and then computes the limit as part of the integration process.